A poset $P$ is said to be connected iff for all $x,y \in P$ we can 'zigzag' from $x$ to $y$ in finitely many steps. That is, iff for all $x,y \in P$ there exists a finite sequence $a : n \rightarrow P$ such that $$x \leq a_0 \geq a_1 \leq \cdots a_{n-2} \geq a_{n-1} \leq y$$ or $$x \leq a_0 \geq a_1 \leq \cdots a_{n-2} \leq a_{n-1} \geq y$$ (depending on whether $n$ is even or odd).
Is there a natural way of endowing a poset $P$ with a topology such that $P$ is connected in the sense defined above iff its topologically connected? (I am not trying to suggest that there's only one way of doing this.)
Lets denote the equivalence relation 'zigzagging' by ~. It partitions the set $P$ into classes $(P_i)$. Taking all possible unions of those classes defines a topology that is connected iff it only contains the empty set and $P$.
This works for any equivalence relation.